Definitions and Basic Properties of Measurable Functions

In this article we introduce some definitions concerning measurable functions and prove related properties. In this paper k is a natural number, r is a real number, i is an integer, and q is a rational number. The subset Z − of R is defined by: (Def. 1) r ∈ Z − iff there exists k such that r = −k. Let us note that Z − is non empty. The following three propositions are true: (1) N ≈ Z −. Z is a subset of R. Let n be a natural number. The functor Q(n) yielding a subset of Q is defined by: (Def. 2) q ∈ Q(n) iff there exists i such that q = i n. Let n be a natural number. One can verify that Q(n + 1) is non empty. We now state two propositions: (4) For every natural number n holds Z ≈ Q(n + 1).